Effect of Hunting on Red Deer

P15.2 Fortgeschrittenes Praxisprojekt

Nikolai German, Thomas Witzani, Ziqi Xu, Zhengchen Yuan, Baisu Zhou

Dr. Nicolas Ferry - Bavarian National Forest Park / Daniel Schlichting - StabLab

31 Jan 2025

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

Motivation

  • Hunting activities have a numerical effect on animal populations
  • Additionally, hunting can have non-lethal effects
  • Goal: assess short-term stress response in red deer towards hunting events at the Bavarian Forest National Park

Data-Generating Process

  • A deer roams freely in the Bavarian Forest National Park
  • Its movement is tracked by a GPS collar
  • A hunting event happens
  • After some time, the deer defecates. The defecation event
  • Subsequently, Researchers go to the defecation location and collect a fecal sample

FCMs as a Measure of Stress

  • Faecal Cortisol Metabolites (FCM) are substances found in feces of animals
  • The FCM level is used to measure previous stress. Higher Stress \(\implies\) Higher FCM level
  • Stress \(\Rightarrow\) secretion of certain hormones \(\Rightarrow\) gut retention \(\Rightarrow\) FCM
  • Gut retention time \(\approx\) 19 hours
  • Once defecated, FCM levels decay over time

Huber et al (2003)

Research Questions

  • What is the effect of temporal and spatial distance on FCM levels?
  • Does the time between defecation event and sample collection effect FCM levels?

Approach

  • Model FCM levels - amongst other covariables - on spatial and temporal distance to hunting activities

  • Expectations:

    • FCM levels higher when closer in time and space
    • FCM levels lower, the more time passes between defecating and sampling

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

The Datasets

  • Movement Data
  • Hunting Events
  • FCM Data

Movement Data

  • Contains the location of the 40 collared deer
  • Period: Feb 2020 - Feb 2023
  • Movement is tracked in hourly intervals

Hunting Events

  • Contains location and date of hunting events
  • Observations: 720 events
  • 532 Observations with complete timestamp

FCM Data

Contains information of 809 faecal samples, including:

  • the FCM level [ng/g]
  • the time and location of sampling
  • to which deer the sample belongs
  • when the defecation happened

Main Challenges

Uncertainty, due to:

  • Hunting Events are reported as single timestamp
  • Location of Deer reported hourly
  • Collared deer are not in the proximity of the reported hunts most of the time
  • Curse of high dimension: Locations of Deer are likely to have a big distance to Hunting Events in one dimension (two spatial, one temporal)

Distance Approximation

Deer location at the time of hunting event is approximated by linear interpolation:

Relevant Hunting Events

To identify relevant Hunting Events respective to a given FCM Sample, we introduce three selection parameters:

  • Gut retention time (GRT) target [hours]: Target Delay between Stress Event and Defecation
  • Gut retention time (GRT) thresholds [hours]: maximum temporal Distance between respective Deer and Hunting Event, with the Minimum beeing Zero
  • Distance threshold [km]: Maximum spatial Distance between respective Deer and potential Hunting Event

The Most Relevant Hunting Event

Among the relevant hunting events, the most relevant one is defined by one of the three introduced proximity criteria:

  • the closest in time to GRT = 19 hours (“closest in time”)
  • the closest in space (“nearest”)
  • the one with the “highest score”

Illustration

TimeDiff Distance 19 hours distance threshold GRT highthreshold Number of otherrelevant huntingevent = 3 Deer Hunting events Nearest Highestscore Closestin time(to 19 hours)

A hunting event is considered relevant to a FCM sample, if

  • the time difference between experiencing stress (hunting) and defecation is between the GRT thresholds, and
  • the distance between the deer and the hunting event is \(\leq\) distance threshold.

The Scoring Function

we define the Scoring function as following:

\[ S(d, t) \propto \begin{cases} \frac{1}{d^2} \cdot f_\textbf{t}(t), t \sim \mathcal{N}(\mu, \sigma^2) &|t \leq \mu \\ \frac{1}{d^2} \cdot f_\textbf{t}(t), t \sim \mathcal{Laplace}(\mu, b) &|t > \mu \end{cases} \] where:

\[ \begin{align*} d & \text{: Distance } \\ t & \text{: Time Difference } \\ \mu & \text{: GRT target = 19 hours } \end{align*} \]

The Scoring Function

The marginal effects of distance and elapsed time since challenge on the score:

The Fused Data

Finish Datasets

We suggest three different Datasets for Modelling

DataSet GRT low GRT high Distance Threshold Proximity Criterion Deers Observations
1 0 36 10 closest in time 35 149
2 0 36 10 nearest 35 147
3 0 200 15 score 36 223

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

The Models

For Modelling, we consider the following covariates, defined for each pair of FCM sample and most relevant hunting event:

  • Time Difference
  • Distance
  • Sample Delay
  • Defecation Day (as Day of Year (1-366))
  • Number of other relevant hunting events

The Models

We chose two different approaches to Modelling:

  1. Machine Learning: a model, which focuses on prediction, in our case a XGBoost Model
  2. Statistical Modelling: a model, which helps to understand the effects of our covariables, here a General Additive Mixed Model

A. XGBoost

TBD

B. Generalized Additive Mixed Model

  • Family: Gamma

  • Log link for interpretability

  • Let \(i = 1,\dots,N\) be the indices of deer and \(j = 1,\dots,n_i\) be the indices of FCM measurements for each deer

\[ \begin{eqnarray} \textup{FCM}_{ij} &\sim& \mathcal{Ga}\left( \nu, \frac{\nu}{\mu_{ij}} \right) \\ \mu_{ij} &=& \mathbb{E}(\textup{FCM}_{ij}) = \exp(\eta_{ij}) \\ \eta_{ij} &=& \beta_0 + \beta_1 \textup{Pregnant}_{ij} + \beta_2 \textup{NumberOtherHunts}_{ij} + \\ && f_1(\textup{TimeDiff}_{ij}) + f_2(\textup{Distance}_{ij}) + \\ && f_3(\textup{SampleDelay}_{ij}) + f_4(\textup{DefecationDay}_{ij}) + \\ && \gamma_{i}, \\ \gamma_i &\overset{\mathrm{iid}}{\sim}& \mathcal{N}(0, \sigma_\gamma^2). \end{eqnarray} \]

B Generalized Additive Mixed Model

Closest in time

Dataset Term Estimate Std_Error
Closest in Time (Intercept) 5.824 0.053
Closest in Time NumOtherHunts -0.137 0.061

B Generalized Additive Mixed Model

nearest

Dataset Term Estimate Std_Error
Nearest (Intercept) 5.812 0.054
Nearest NumOtherHunts -0.103 0.060

B Generalized Additive Mixed Model

Highest score

Dataset Term Estimate Std_Error
Highest Score (Intercept) 5.888 0.081
Highest Score NumOtherHunts -0.011 0.014

B Results

Category Subcategory Description
Diagnostics QQ Plot Residuals mostly follow expected distribution
Diagnostics Residuals vs Predictor No major pattern
Diagnostics Histogram Reasonable fit, some variance
Diagnostics Observed vs Fitted Moderate spread, some unexplained variance
Random Effects Time & Space Effects Weak or inconsistent
Random Effects Sample Delay Shows some effect
Linear Effects other hunting events No significant impact

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

Conclusion

  • Due to the high uncertainties, we were not able to detect a relevant effect of spatial or temporal distance on FCM levels
  • In some of the cases we were able to prove the expected decay of FCM levels with prolonged time between defecation event and sample collection
  • With more datapoints, the uncertainty will likely shrink

Discussion

  • How to minimize spatial and temporal distance at the same time?

  • How to use a bigger Part of the Data?

Effect of Hunting on Red Deer